
 The modular logarithmic tropicalization map \(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}:J_{g,n}^{log}\rightarrow \widetilde{\mathcal {J}}^{trop}_{g,n} \)  is strict, smooth and surjective.

 The morphism of Artin fans
\(\mathcal {A}(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}):\mathcal {A}_{J_{g,n}^{log}}\rightarrow \mathcal {A}_{\widetilde{\mathcal {J}}^{trop}_{g,n}}\) 
is an isomorphism. In particular, we have a commutative diagram
\({J_{g,n}^{log} [rr]^{\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}} [d]_{\operatorname{trop}_{J_{g,n}^{log}}}&&\widetilde{\mathcal {J}}^{trop}_{g,n} [d]^{\operatorname{trop}_{\widetilde{\mathcal {J}}^{trop}_{g,n}}} \\\mathcal {A}_{J_{g,n}^{log}} [rr]_{\mathcal {A}(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}})}^{\cong } && \mathcal {A}_{\widetilde{\mathcal {J}}^{trop}_{g,n}}}\) 


There exist morphisms of Artin fans \(\mathcal {A}(\widetilde{\Phi }^{trop}): \mathcal {A}_{\widetilde{\mathcal {J}}^{trop}_{g,n}}\rightarrow \mathcal {A}_{\widetilde{\mathcal {M}}^{trop}_{g,n}}\)  and \(\mathcal {A}(\Phi ^{log}): \mathcal {A}_{J_{g,n}^{log}}\rightarrow \mathcal {A}_{\mathcal {M}_{g,n}^{log}}\)  sitting into the commutative diagram
\({J_{g,n}^{log}[rr]_{\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}} @/^2pc/[rrrrrr]^{\operatorname{trop}_{J_{g,n}^{log}}} [d]^{\Phi ^{log}} && \widetilde{\mathcal {J}}^{trop}_{g,n} [rr]_{\operatorname{trop}_{\widetilde{\mathcal {J}}^{trop}_{g,n}}} [d]^{\widetilde{\Phi }^{trop}}&& \mathcal {A}_{\widetilde{\mathcal {J}}^{trop}_{g,n}} [d]^{\mathcal {A}(\widetilde{\Phi }^{trop})}&& [ll]_{\cong }^{\mathcal {A}(\widetilde{\operatorname{trop}}_{J_{g,n}^{log}})} \mathcal {A}_{J_{g,n}^{log}} [d]^{\mathcal {A}(\Phi ^{log})},\\\mathcal {M}_{g,n}^{log}[rr]^{\widetilde{\operatorname{trop}}_{\mathcal {M}_{g,n}^{log}}} @/_2pc/[rrrrrr]_{\operatorname{trop}_{J_{g,n}^{log}}} && \widetilde{\mathcal {M}}^{trop}_{g,n} [rr]^{\operatorname{trop}_{\widetilde{\mathcal {M}}^{trop}_{g,n}}} && \mathcal {A}_{\widetilde{\mathcal {M}}^{trop}_{g,n}} && [ll]^{\cong }_{\mathcal {A}(\widetilde{\operatorname{trop}}_{\mathcal {M}_{g,n}^{log}})} \mathcal {A}_{\mathcal {M}_{g,n}^{log}},\\}\) 
where the first row is the diagram (REF ) and the second row is the analogous diagram established in [1]} (building upon [2]}).

 The two morphisms of logarithmic algebraic stacks
\({\overline{\mathcal {J}}_{g,n}&& J_{g,n}^{log} [ll]_{\Upsilon _{J_{g,n}^{log}}} [rr]^{\widetilde{\operatorname{trop}}_{J_{g,n}^{log}}} && \widetilde{\mathcal {J}}^{trop}_{g,n}}\) 
are compatible with the stratifications (REF ) and (REF ), i.e.
\((\widetilde{\operatorname{trop}}_{J_{g,n}^{log}})^{-1}(\widetilde{\mathcal {J}}^{trop}_{(G,D)})=(\Upsilon _{J_{g,n}^{log}})^{-1}(J_{(G,D)})\) 
for every \((G,D)\in \operatorname{\mathcal {QD}iv}_{g,n}\) .
In particular, we have that

\((\widetilde{\operatorname{trop}}_{J_{g,n}^{log}})^{-1}(\widetilde{\mathcal {J}}^{trop}_{g,n,d})=J_{g,n,d}^{log}=(\Upsilon _{J_{g,n}^{log}})^{-1}(\overline{\mathcal {J}}_{g,n,d})\)  for any \(d\in \mathbb {Z}\) .

\((\widetilde{\operatorname{trop}}_{J_{g,n}^{log}})^{-1}(\widetilde{\mathcal {J}}_{g,n,(d)}^{trop. \operatorname{spl}})=J_{g,n,(d)}^{log,spl}=(\Upsilon _{J_{g,n}^{log}})^{-1}(\overline{\mathcal {J}}_{g,n,(d)}^{\operatorname{spl}})\) .

\((\widetilde{\operatorname{trop}}_{J_{g,n}^{log}})^{-1}(\widetilde{\mathcal {J}}^{trop}_{g,n}(\phi ))= \mathcal {J}_{g,n}^{log}(\phi )=(\Upsilon _{J_{g,n}^{log}})^{-1}(\overline{\mathcal {J}}_{g,n}(\phi ))\)  for any universal stability condition \(\phi \in V_{g,n}\) .



